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Manuscript/eDFT.tex
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@ -121,7 +121,9 @@
\begin{document} \begin{document}
\title{Weight-dependent local density-functional approximations for ensembles} \title{Weight-dependent local density-functional \manu{approximation to
ensemble correlation energies}}
%\title{Weight-dependent local density-functional approximations for ensembles}
\author{Pierre-Fran\c{c}ois Loos} \author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
@ -1272,10 +1274,11 @@ drastically.
%correlation is strong. It is not clear to me which integral ($W_{01}?$) %correlation is strong. It is not clear to me which integral ($W_{01}?$)
%drives the all thing.\\} %drives the all thing.\\}
It is important to note that, even though the GIC removes the explicit It is important to note that, even though the GIC removes the explicit
quadratic terms from the ensemble energy, a non-negligible curvature quadratic \manu{Hx} terms from the ensemble energy, a non-negligible curvature
remains in the GIC-eLDA ensemble energy. \manu{This might be due to remains in the GIC-eLDA ensemble energy \manu{when the electron
correlation is strong}. \manu{This is due to
\textit{(i)} the correlation eLDA \textit{(i)} the correlation eLDA
functional, which induces linear or quadratic weight dependencies of the individual functional, which contributes linearly (or even quadratically) to the individual
energies [see Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and energies [see Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
\eqref{eq:Taylor_exp_DDisc_term}], and \textit{(ii)} the optimization of the \eqref{eq:Taylor_exp_DDisc_term}], and \textit{(ii)} the optimization of the
ensemble KS orbitals in the presence of ghost-interaction errors {[see ensemble KS orbitals in the presence of ghost-interaction errors {[see
@ -1317,10 +1320,10 @@ the ground and first excited-state increase with respect to the
first-excited-state weight $\ew{1}$, thus showing that, in this first-excited-state weight $\ew{1}$, thus showing that, in this
case, we case, we
``deteriorate'' these states by optimizing the orbitals for the ``deteriorate'' these states by optimizing the orbitals for the
ensemble, rather than for each state individually. The reverse actually occurs for the ground state in the triensemble ensemble, rather than for each state separately. The reverse actually occurs for the ground state in the triensemble
as $\ew{2}$ increases. The variations in the ensemble as $\ew{2}$ increases. The variations in the ensemble
weights are essentially linear or quadratic. They are induced by the weights are essentially linear or quadratic. They are induced by the
eLDA functional, as readily seen from eLDA correlation functional, as readily seen from
Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first \eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
@ -1445,7 +1448,7 @@ correlation ensemble derivative contribution $\DD{c}{(I)}$
to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}]. to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
In our case, both single ($I=1$) and double ($I=2$) excitations are considered. In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, for $\nEl = 3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$ To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, for $\nEl = 3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$
on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}]. on the excitation energies obtained at the KS-eLDA level with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
%\manu{Manu: there is something I do not understand. If you want to %\manu{Manu: there is something I do not understand. If you want to
%evaluate the importance of the ensemble correlation derivatives you %evaluate the importance of the ensemble correlation derivatives you
%should only remove the following contribution from the $K$th KS-eLDA %should only remove the following contribution from the $K$th KS-eLDA
@ -1457,13 +1460,12 @@ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}
%%rather than $E^{(I)}_{\rm HF}$ %%rather than $E^{(I)}_{\rm HF}$
%} %}
We first stress that although for $\nEl=3$ both single and double excitation energies are We first stress that although for $\nEl=3$ both single and double excitation energies are
systematically improved, as the strength of electron correlation systematically improved (as the strength of electron correlation
increases, when increases) when
taking into account taking into account
the correlation ensemble derivative, this is not the correlation ensemble derivative, this is not
always the case for larger numbers of electrons. always the case for larger numbers of electrons.
The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime. For 3-boxium, in the zero-weight limit, the ensemble derivative is
For 3-boxium, in the zero-weight limit, its contribution is
significantly larger for the single significantly larger for the single
excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
case. case.